LMFDB - Embedded newform 325.6.b.h.274.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,6,Mod(274,325)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("325.274");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	325.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$52.1247414392$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} + 378 x^{16} + 59175 x^{14} + 4956036 x^{12} + 239061247 x^{10} + 6629044458 x^{8} + \cdots + 134659641600 $ x^18 + 378x^16 + 59175x^14 + 4956036x^12 + 239061247x^10 + 6629044458x^8 + 98240243849x^6 + 625586966104x^4 + 584919029904x^2 + 134659641600
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.5
Root			$-8.28147i$ of defining polynomial
Character	$\chi$	$=$	325.274
Dual form			325.6.b.h.274.14

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.28147i q^{2} -14.9935i q^{3} -21.0198 q^{4} -109.175 q^{6} -135.391i q^{7} -79.9521i q^{8} +18.1955 q^{9} +O(q^{10})$	\(q-7.28147i q^{2} -14.9935i q^{3} -21.0198 q^{4} -109.175 q^{6} -135.391i q^{7} -79.9521i q^{8} +18.1955 q^{9} +191.317 q^{11} +315.160i q^{12} +169.000i q^{13} -985.848 q^{14} -1254.80 q^{16} +874.338i q^{17} -132.490i q^{18} -1992.26 q^{19} -2029.99 q^{21} -1393.07i q^{22} -2091.74i q^{23} -1198.76 q^{24} +1230.57 q^{26} -3916.23i q^{27} +2845.90i q^{28} +46.3621 q^{29} -9450.42 q^{31} +6578.33i q^{32} -2868.52i q^{33} +6366.46 q^{34} -382.466 q^{36} +3374.82i q^{37} +14506.6i q^{38} +2533.90 q^{39} -10511.6 q^{41} +14781.3i q^{42} +4056.27i q^{43} -4021.45 q^{44} -15230.9 q^{46} +17035.2i q^{47} +18813.8i q^{48} -1523.82 q^{49} +13109.4 q^{51} -3552.34i q^{52} -23984.5i q^{53} -28515.9 q^{54} -10824.8 q^{56} +29870.9i q^{57} -337.584i q^{58} +10197.8 q^{59} -27440.0 q^{61} +68812.9i q^{62} -2463.52i q^{63} +7746.27 q^{64} -20887.0 q^{66} +317.851i q^{67} -18378.4i q^{68} -31362.5 q^{69} +39664.7 q^{71} -1454.77i q^{72} +4764.06i q^{73} +24573.7 q^{74} +41876.9 q^{76} -25902.7i q^{77} -18450.5i q^{78} +74545.9 q^{79} -54296.4 q^{81} +76539.5i q^{82} -14234.5i q^{83} +42669.9 q^{84} +29535.6 q^{86} -695.130i q^{87} -15296.2i q^{88} +132052. q^{89} +22881.1 q^{91} +43967.9i q^{92} +141695. i q^{93} +124041. q^{94} +98632.1 q^{96} -19366.7i q^{97} +11095.6i q^{98} +3481.12 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9}+O(q^{10}) $ 18 * q - 182 * q^4 - 166 * q^6 - 1124 * q^9	$ 18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9} - 2844 q^{11} + 684 q^{14} - 2122 q^{16} + 816 q^{19} - 7824 q^{21} + 16938 q^{24} - 1690 q^{26} + 17474 q^{29} + 1496 q^{31} + 35578 q^{34} + 1024 q^{36} + 3718 q^{39} - 57352 q^{41} + 61198 q^{44} - 24112 q^{46} + 81814 q^{49} - 62012 q^{51} + 205522 q^{54} - 46004 q^{56} + 176284 q^{59} + 56330 q^{61} + 201690 q^{64} + 85154 q^{66} + 363494 q^{69} - 141124 q^{71} + 271352 q^{74} + 92746 q^{76} + 328146 q^{79} - 139870 q^{81} + 691312 q^{84} - 589840 q^{86} + 505396 q^{89} + 4056 q^{91} + 1003212 q^{94} - 638362 q^{96} + 1515552 q^{99}+O(q^{100}) $ 18 * q - 182 * q^4 - 166 * q^6 - 1124 * q^9 - 2844 * q^11 + 684 * q^14 - 2122 * q^16 + 816 * q^19 - 7824 * q^21 + 16938 * q^24 - 1690 * q^26 + 17474 * q^29 + 1496 * q^31 + 35578 * q^34 + 1024 * q^36 + 3718 * q^39 - 57352 * q^41 + 61198 * q^44 - 24112 * q^46 + 81814 * q^49 - 62012 * q^51 + 205522 * q^54 - 46004 * q^56 + 176284 * q^59 + 56330 * q^61 + 201690 * q^64 + 85154 * q^66 + 363494 * q^69 - 141124 * q^71 + 271352 * q^74 + 92746 * q^76 + 328146 * q^79 - 139870 * q^81 + 691312 * q^84 - 589840 * q^86 + 505396 * q^89 + 4056 * q^91 + 1003212 * q^94 - 638362 * q^96 + 1515552 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 7.28147i	− 1.28719i	−0.765365	−	0.643597i	$-0.777441\pi$
$2$	− 7.28147i	− 1.28719i	0.765365	−	0.643597i	$-0.222559\pi$
$3$	− 14.9935i	− 0.961832i	−0.876767	−	0.480916i	$-0.840304\pi$
$3$	− 14.9935i	− 0.961832i	0.876767	−	0.480916i	$-0.159696\pi$
$4$	−21.0198	−0.656868
$5$	0	0
$5$	0	0
$6$	−109.175	−1.23806
$7$	− 135.391i	− 1.04435i	−0.852839	−	0.522175i	$-0.825121\pi$
$7$	− 135.391i	− 1.04435i	0.852839	−	0.522175i	$-0.174879\pi$
$8$	− 79.9521i	− 0.441677i
$9$	18.1955	0.0748787
$10$	0	0
$11$	191.317	0.476731	0.238365	−	0.971176i	$-0.423388\pi$
$11$	191.317	0.476731	0.238365	+	0.971176i	$0.423388\pi$
$12$	315.160i	0.631797i
$13$	169.000i	0.277350i
$13$	169.000i	0.277350i
$14$	−985.848	−1.34428
$15$	0	0
$16$	−1254.80	−1.22539
$17$	874.338i	0.733765i	0.930267	+	0.366883i	$0.119575\pi$
$17$	874.338i	0.733765i	−0.930267	+	0.366883i	$0.880425\pi$
$18$	− 132.490i	− 0.0963834i
$19$	−1992.26	−1.26608	−0.633042	−	0.774118i	$-0.718194\pi$
$19$	−1992.26	−1.26608	−0.633042	+	0.774118i	$0.718194\pi$
$20$	0	0
$21$	−2029.99	−1.00449
$22$	− 1393.07i	− 0.613645i
$23$	− 2091.74i	− 0.824495i	−0.911072	−	0.412248i	$-0.864744\pi$
$23$	− 2091.74i	− 0.824495i	0.911072	−	0.412248i	$-0.135256\pi$
$24$	−1198.76	−0.424819
$25$	0	0
$26$	1230.57	0.357003
$27$	− 3916.23i	− 1.03385i
$28$	2845.90i	0.686000i
$29$	46.3621	0.0102369	0.00511845	−	0.999987i	$-0.498371\pi$
$29$	46.3621	0.0102369	0.00511845	+	0.999987i	$0.498371\pi$
$30$	0	0
$31$	−9450.42	−1.76623	−0.883114	−	0.469158i	$-0.844558\pi$
$31$	−9450.42	−1.76623	−0.883114	+	0.469158i	$0.844558\pi$
$32$	6578.33i	1.13564i
$33$	− 2868.52i	− 0.458535i
$34$	6366.46	0.944498
$35$	0	0
$36$	−382.466	−0.0491854
$37$	3374.82i	0.405272i	0.979254	+	0.202636i	$0.0649508\pi$
$37$	3374.82i	0.405272i	−0.979254	+	0.202636i	$0.935049\pi$
$38$	14506.6i	1.62969i
$39$	2533.90	0.266764
$40$	0	0
$41$	−10511.6	−0.976578	−0.488289	−	0.872682i	$-0.662379\pi$
$41$	−10511.6	−0.976578	−0.488289	+	0.872682i	$0.662379\pi$
$42$	14781.3i	1.29297i
$43$	4056.27i	0.334546i	0.985911	+	0.167273i	$0.0534961\pi$
$43$	4056.27i	0.334546i	−0.985911	+	0.167273i	$0.946504\pi$
$44$	−4021.45	−0.313149
$45$	0	0
$46$	−15230.9	−1.06129
$47$	17035.2i	1.12487i	0.826842	+	0.562435i	$0.190135\pi$
$47$	17035.2i	1.12487i	−0.826842	+	0.562435i	$0.809865\pi$
$48$	18813.8i	1.17862i
$49$	−1523.82	−0.0906658
$50$	0	0
$51$	13109.4	0.705759
$52$	− 3552.34i	− 0.182182i
$53$	− 23984.5i	− 1.17285i	−0.810004	−	0.586424i	$-0.800535\pi$
$53$	− 23984.5i	− 1.17285i	0.810004	−	0.586424i	$-0.199465\pi$
$54$	−28515.9	−1.33077
$55$	0	0
$56$	−10824.8	−0.461265
$57$	29870.9i	1.21776i
$58$	− 337.584i	− 0.0131769i
$59$	10197.8	0.381398	0.190699	−	0.981649i	$-0.438925\pi$
$59$	10197.8	0.381398	0.190699	+	0.981649i	$0.438925\pi$
$60$	0	0
$61$	−27440.0	−0.944191	−0.472095	−	0.881547i	$-0.656502\pi$
$61$	−27440.0	−0.944191	−0.472095	+	0.881547i	$0.656502\pi$
$62$	68812.9i	2.27348i
$63$	− 2463.52i	− 0.0781995i
$64$	7746.27	0.236397
$65$	0	0
$66$	−20887.0	−0.590223
$67$	317.851i	0.00865040i	0.999991	+	0.00432520i	$0.00137676\pi$
$67$	317.851i	0.00865040i	−0.999991	+	0.00432520i	$0.998623\pi$
$68$	− 18378.4i	− 0.481987i
$69$	−31362.5	−0.793026
$70$	0	0
$71$	39664.7	0.933809	0.466905	−	0.884308i	$-0.345369\pi$
$71$	39664.7	0.933809	0.466905	+	0.884308i	$0.345369\pi$
$72$	− 1454.77i	− 0.0330722i
$73$	4764.06i	0.104633i	0.998631	+	0.0523167i	$0.0166605\pi$
$73$	4764.06i	0.104633i	−0.998631	+	0.0523167i	$0.983339\pi$
$74$	24573.7	0.521664
$75$	0	0
$76$	41876.9	0.831650
$77$	− 25902.7i	− 0.497873i
$78$	− 18450.5i	− 0.343377i
$79$	74545.9	1.34387	0.671933	−	0.740612i	$-0.265464\pi$
$79$	74545.9	1.34387	0.671933	+	0.740612i	$0.265464\pi$
$80$	0	0
$81$	−54296.4	−0.919514
$82$	76539.5i	1.25705i
$83$	− 14234.5i	− 0.226803i	−0.993549	−	0.113401i	$-0.963825\pi$
$83$	− 14234.5i	− 0.226803i	0.993549	−	0.113401i	$-0.0361746\pi$
$84$	42669.9	0.659817
$85$	0	0
$86$	29535.6	0.430625
$87$	− 695.130i	− 0.00984618i
$88$	− 15296.2i	− 0.210561i
$89$	132052.	1.76713	0.883566	−	0.468307i	$-0.155136\pi$
$89$	132052.	1.76713	0.883566	+	0.468307i	$0.155136\pi$
$90$	0	0
$91$	22881.1	0.289650
$92$	43967.9i	0.541585i
$93$	141695.i	1.69882i
$94$	124041.	1.44793
$95$	0	0
$96$	98632.1	1.09230
$97$	− 19366.7i	− 0.208990i	−0.994525	−	0.104495i	$-0.966677\pi$
$97$	− 19366.7i	− 0.208990i	0.994525	−	0.104495i	$-0.0333226\pi$
$98$	11095.6i	0.116704i
$99$	3481.12	0.0356970
$100$	0	0
$101$	−90391.7	−0.881709	−0.440855	−	0.897579i	$-0.645325\pi$
$101$	−90391.7	−0.881709	−0.440855	+	0.897579i	$0.645325\pi$
$102$	− 95455.5i	− 0.908449i
$103$	59762.2i	0.555051i	0.960718	+	0.277526i	$0.0895144\pi$
$103$	59762.2i	0.555051i	−0.960718	+	0.277526i	$0.910486\pi$
$104$	13511.9	0.122499
$105$	0	0
$106$	−174643.	−1.50968
$107$	− 178407.i	− 1.50645i	−0.657766	−	0.753223i	$-0.728498\pi$
$107$	− 178407.i	− 1.50645i	0.657766	−	0.753223i	$-0.271502\pi$
$108$	82318.3i	0.679105i
$109$	37689.5	0.303847	0.151923	−	0.988392i	$-0.451453\pi$
$109$	37689.5	0.303847	0.151923	+	0.988392i	$0.451453\pi$
$110$	0	0
$111$	50600.3	0.389804
$112$	169889.i	1.27974i
$113$	− 38616.5i	− 0.284497i	−0.989831	−	0.142248i	$-0.954567\pi$
$113$	− 38616.5i	− 0.284497i	0.989831	−	0.142248i	$-0.0454331\pi$
$114$	217504.	1.56749
$115$	0	0
$116$	−974.522	−0.00672429
$117$	3075.04i	0.0207676i
$118$	− 74255.3i	− 0.490933i
$119$	118378.	0.766307
$120$	0	0
$121$	−124449.	−0.772728
$122$	199804.i	1.21536i
$123$	157605.i	0.939305i
$124$	198646.	1.16018
$125$	0	0
$126$	−17938.0	−0.100658
$127$	− 301119.i	− 1.65664i	−0.560255	−	0.828320i	$-0.689297\pi$
$127$	− 301119.i	− 1.65664i	0.560255	−	0.828320i	$-0.310703\pi$
$128$	154102.i	0.831351i
$129$	60817.6	0.321777
$130$	0	0
$131$	−100245.	−0.510371	−0.255186	−	0.966892i	$-0.582137\pi$
$131$	−100245.	−0.510371	−0.255186	+	0.966892i	$0.582137\pi$
$132$	60295.6i	0.301197i
$133$	269735.i	1.32223i
$134$	2314.42	0.0111347
$135$	0	0
$136$	69905.2	0.324087
$137$	− 72376.9i	− 0.329457i	−0.986339	−	0.164728i	$-0.947325\pi$
$137$	− 72376.9i	− 0.329457i	0.986339	−	0.164728i	$-0.0526748\pi$
$138$	228365.i	1.02078i
$139$	−296858.	−1.30320	−0.651601	−	0.758562i	$-0.725902\pi$
$139$	−296858.	−1.30320	−0.651601	+	0.758562i	$0.725902\pi$
$140$	0	0
$141$	255417.	1.08194
$142$	− 288817.i	− 1.20199i
$143$	32332.7i	0.132221i
$144$	−22831.8	−0.0917558
$145$	0	0
$146$	34689.3	0.134683
$147$	22847.4i	0.0872053i
$148$	− 70938.0i	− 0.266210i
$149$	−16259.4	−0.0599981	−0.0299991	−	0.999550i	$-0.509550\pi$
$149$	−16259.4	−0.0599981	−0.0299991	+	0.999550i	$0.509550\pi$
$150$	0	0
$151$	383495.	1.36873	0.684364	−	0.729140i	$-0.260080\pi$
$151$	383495.	1.36873	0.684364	+	0.729140i	$0.260080\pi$
$152$	159285.i	0.559200i
$153$	15909.0i	0.0549434i
$154$	−188610.	−0.640859
$155$	0	0
$156$	−53262.0	−0.175229
$157$	− 235502.i	− 0.762510i	−0.924470	−	0.381255i	$-0.875492\pi$
$157$	− 235502.i	− 0.762510i	0.924470	−	0.381255i	$-0.124508\pi$
$158$	− 542804.i	− 1.72982i
$159$	−359612.	−1.12808
$160$	0	0
$161$	−283204.	−0.861061
$162$	395358.i	1.18359i
$163$	205165.i	0.604832i	0.953176	+	0.302416i	$0.0977932\pi$
$163$	205165.i	0.604832i	−0.953176	+	0.302416i	$0.902207\pi$
$164$	220951.	0.641483
$165$	0	0
$166$	−103648.	−0.291939
$167$	− 68965.6i	− 0.191356i	−0.995412	−	0.0956778i	$-0.969498\pi$
$167$	− 68965.6i	− 0.191356i	0.995412	−	0.0956778i	$-0.0305018\pi$
$168$	162302.i	0.443660i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	−36250.2	−0.0948027
$172$	− 85261.9i	− 0.219753i
$173$	613056.i	1.55735i	0.627430	+	0.778673i	$0.284107\pi$
$173$	613056.i	1.55735i	−0.627430	+	0.778673i	$0.715893\pi$
$174$	−5061.56	−0.0126739
$175$	0	0
$176$	−240066.	−0.584182
$177$	− 152901.i	− 0.366841i
$178$	− 961530.i	− 2.27464i
$179$	637734.	1.48767	0.743836	−	0.668362i	$-0.233004\pi$
$179$	637734.	1.48767	0.743836	+	0.668362i	$0.233004\pi$
$180$	0	0
$181$	145338.	0.329749	0.164875	−	0.986315i	$-0.447278\pi$
$181$	145338.	0.329749	0.164875	+	0.986315i	$0.447278\pi$
$182$	− 166608.i	− 0.372836i
$183$	411421.i	0.908153i
$184$	−167239.	−0.364161
$185$	0	0
$186$	1.03175e6	2.18671
$187$	167276.i	0.349808i
$188$	− 358076.i	− 0.738891i
$189$	−530224.	−1.07970
$190$	0	0
$191$	−967379.	−1.91873	−0.959364	−	0.282173i	$-0.908945\pi$
$191$	−967379.	−1.91873	−0.959364	+	0.282173i	$0.908945\pi$
$192$	− 116143.i	− 0.227375i
$193$	− 9104.15i	− 0.0175932i	−0.999961	−	0.00879662i	$-0.997200\pi$
$193$	− 9104.15i	− 0.0175932i	0.999961	−	0.00879662i	$-0.00280009\pi$
$194$	−141018.	−0.269011
$195$	0	0
$196$	32030.4	0.0595555
$197$	567218.i	1.04132i	0.853764	+	0.520660i	$0.174314\pi$
$197$	567218.i	1.04132i	−0.853764	+	0.520660i	$0.825686\pi$
$198$	− 25347.7i	− 0.0459489i
$199$	321297.	0.575141	0.287571	−	0.957759i	$-0.407152\pi$
$199$	321297.	0.575141	0.287571	+	0.957759i	$0.407152\pi$
$200$	0	0
$201$	4765.69	0.00832023
$202$	658185.i	1.13493i
$203$	− 6277.03i	− 0.0106909i
$204$	−275556.	−0.463591
$205$	0	0
$206$	435156.	0.714459
$207$	− 38060.3i	− 0.0617371i
$208$	− 212061.i	− 0.339863i
$209$	−381154.	−0.603580
$210$	0	0
$211$	−165479.	−0.255880	−0.127940	−	0.991782i	$-0.540836\pi$
$211$	−165479.	−0.255880	−0.127940	+	0.991782i	$0.540836\pi$
$212$	504150.i	0.770407i
$213$	− 594712.i	− 0.898168i
$214$	−1.29907e6	−1.93909
$215$	0	0
$216$	−313111.	−0.456629
$217$	1.27951e6i	1.84456i
$218$	− 274435.i	− 0.391109i
$219$	71429.8	0.100640
$220$	0	0
$221$	−147763.	−0.203510
$222$	− 368445.i	− 0.501753i
$223$	− 1.14125e6i	− 1.53680i	−0.639968	−	0.768402i	$-0.721052\pi$
$223$	− 1.14125e6i	− 1.53680i	0.639968	−	0.768402i	$-0.278948\pi$
$224$	890649.	1.18601
$225$	0	0
$226$	−281185.	−0.366202
$227$	− 573565.i	− 0.738785i	−0.929273	−	0.369393i	$-0.879566\pi$
$227$	− 573565.i	− 0.738785i	0.929273	−	0.369393i	$-0.120434\pi$
$228$	− 627880.i	− 0.799908i
$229$	−1.11647e6	−1.40688	−0.703440	−	0.710755i	$-0.748353\pi$
$229$	−1.11647e6	−1.40688	−0.703440	+	0.710755i	$0.748353\pi$
$230$	0	0
$231$	−388372.	−0.478871
$232$	− 3706.75i	− 0.00452140i
$233$	− 1.08552e6i	− 1.30993i	−0.755658	−	0.654967i	$-0.772683\pi$
$233$	− 1.08552e6i	− 1.30993i	0.755658	−	0.654967i	$-0.227317\pi$
$234$	22390.8	0.0267319
$235$	0	0
$236$	−214356.	−0.250528
$237$	− 1.11770e6i	− 1.29257i
$238$	− 861964.i	− 0.986386i
$239$	−497543.	−0.563425	−0.281712	−	0.959499i	$-0.590902\pi$
$239$	−497543.	−0.563425	−0.281712	+	0.959499i	$0.590902\pi$
$240$	0	0
$241$	−1.68881e6	−1.87300	−0.936499	−	0.350669i	$-0.885954\pi$
$241$	−1.68881e6	−1.87300	−0.936499	+	0.350669i	$0.885954\pi$
$242$	906169.i	0.994651i
$243$	− 137552.i	− 0.149434i
$244$	576783.	0.620209
$245$	0	0
$246$	1.14759e6	1.20907
$247$	− 336692.i	− 0.351148i
$248$	755581.i	0.780103i
$249$	−213425.	−0.218146
$250$	0	0
$251$	−358692.	−0.359366	−0.179683	−	0.983725i	$-0.557507\pi$
$251$	−358692.	−0.359366	−0.179683	+	0.983725i	$0.557507\pi$
$252$	51782.6i	0.0513668i
$253$	− 400187.i	− 0.393062i
$254$	−2.19259e6	−2.13242
$255$	0	0
$256$	1.36997e6	1.30651
$257$	− 715783.i	− 0.676003i	−0.941146	−	0.338001i	$-0.890249\pi$
$257$	− 715783.i	− 0.676003i	0.941146	−	0.338001i	$-0.109751\pi$
$258$	− 442841.i	− 0.414189i
$259$	456922.	0.423246
$260$	0	0
$261$	843.583	0.000766525 0
$262$	729933.i	0.656946i
$263$	167988.i	0.149757i	0.997193	+	0.0748787i	$0.0238569\pi$
$263$	167988.i	0.149757i	−0.997193	+	0.0748787i	$0.976143\pi$
$264$	−229344.	−0.202524
$265$	0	0
$266$	1.96407e6	1.70197
$267$	− 1.97991e6i	− 1.69968i
$268$	− 6681.15i	− 0.00568217i
$269$	−1.93165e6	−1.62760	−0.813800	−	0.581145i	$-0.802605\pi$
$269$	−1.93165e6	−1.62760	−0.813800	+	0.581145i	$0.802605\pi$
$270$	0	0
$271$	2.04198e6	1.68900	0.844499	−	0.535558i	$-0.179899\pi$
$271$	2.04198e6	1.68900	0.844499	+	0.535558i	$0.179899\pi$
$272$	− 1.09712e6i	− 0.899150i
$273$	− 343068.i	− 0.278595i
$274$	−527010.	−0.424075
$275$	0	0
$276$	659232.	0.520914
$277$	− 1.05919e6i	− 0.829416i	−0.909955	−	0.414708i	$-0.863884\pi$
$277$	− 1.05919e6i	− 0.829416i	0.909955	−	0.414708i	$-0.136116\pi$
$278$	2.16156e6i	1.67747i
$279$	−171955.	−0.132253
$280$	0	0
$281$	−935578.	−0.706829	−0.353414	−	0.935467i	$-0.614979\pi$
$281$	−935578.	−0.706829	−0.353414	+	0.935467i	$0.614979\pi$
$282$	− 1.85981e6i	− 1.39266i
$283$	− 827252.i	− 0.614005i	−0.951709	−	0.307002i	$-0.900674\pi$
$283$	− 827252.i	− 0.614005i	0.951709	−	0.307002i	$-0.0993260\pi$
$284$	−833743.	−0.613390
$285$	0	0
$286$	235429.	0.170194
$287$	1.42317e6i	1.01989i
$288$	119696.i	0.0850353i
$289$	655390.	0.461589
$290$	0	0
$291$	−290374.	−0.201013
$292$	− 100139.i	− 0.0687303i
$293$	1.77525e6i	1.20806i	0.796960	+	0.604032i	$0.206440\pi$
$293$	1.77525e6i	1.20806i	−0.796960	+	0.604032i	$0.793560\pi$
$294$	166362.	0.112250
$295$	0	0
$296$	269824.	0.178999
$297$	− 749243.i	− 0.492869i
$298$	118392.i	0.0772292i
$299$	353504.	0.228674
$300$	0	0
$301$	549184.	0.349383
$302$	− 2.79241e6i	− 1.76182i
$303$	1.35529e6i	0.848056i
$304$	2.49989e6	1.55145
$305$	0	0
$306$	115841.	0.0707228
$307$	293202.i	0.177550i	0.996052	+	0.0887750i	$0.0282952\pi$
$307$	293202.i	0.177550i	−0.996052	+	0.0887750i	$0.971705\pi$
$308$	544470.i	0.327037i
$309$	896043.	0.533866
$310$	0	0
$311$	−2.34227e6	−1.37321	−0.686605	−	0.727031i	$-0.740900\pi$
$311$	−2.34227e6	−1.37321	−0.686605	+	0.727031i	$0.740900\pi$
$312$	− 202590.i	− 0.117824i
$313$	1.20634e6i	0.695997i	0.937495	+	0.347999i	$0.113139\pi$
$313$	1.20634e6i	0.695997i	−0.937495	+	0.347999i	$0.886861\pi$
$314$	−1.71480e6	−0.981499
$315$	0	0
$316$	−1.56694e6	−0.882743
$317$	2.21969e6i	1.24063i	0.784352	+	0.620316i	$0.212996\pi$
$317$	2.21969e6i	1.24063i	−0.784352	+	0.620316i	$0.787004\pi$
$318$	2.61850e6i	1.45206i
$319$	8869.88	0.00488024
$320$	0	0
$321$	−2.67495e6	−1.44895
$322$	2.06214e6i	1.10835i
$323$	− 1.74191e6i	− 0.929008i
$324$	1.14130e6	0.604000
$325$	0	0
$326$	1.49390e6	0.778536
$327$	− 565097.i	− 0.292249i
$328$	840420.i	0.431332i
$329$	2.30642e6	1.17476
$330$	0	0
$331$	1.37355e6	0.689087	0.344544	−	0.938770i	$-0.388034\pi$
$331$	1.37355e6	0.689087	0.344544	+	0.938770i	$0.388034\pi$
$332$	299207.i	0.148980i
$333$	61406.7i	0.0303462i
$334$	−502171.	−0.246312
$335$	0	0
$336$	2.54723e6	1.23089
$337$	− 428681.i	− 0.205617i	−0.994701	−	0.102809i	$-0.967217\pi$
$337$	− 428681.i	− 0.205617i	0.994701	−	0.102809i	$-0.0327829\pi$
$338$	207966.i	0.0990149i
$339$	−578996.	−0.273638
$340$	0	0
$341$	−1.80803e6	−0.842015
$342$	263955.i	0.122029i
$343$	− 2.06921e6i	− 0.949663i
$344$	324307.	0.147761
$345$	0	0
$346$	4.46395e6	2.00461
$347$	− 3.34049e6i	− 1.48932i	−0.667446	−	0.744658i	$-0.732612\pi$
$347$	− 3.34049e6i	− 1.48932i	0.667446	−	0.744658i	$-0.267388\pi$
$348$	14611.5i	0.00646764i
$349$	1.65843e6	0.728840	0.364420	−	0.931235i	$-0.381267\pi$
$349$	1.65843e6	0.728840	0.364420	+	0.931235i	$0.381267\pi$
$350$	0	0
$351$	661843.	0.286739
$352$	1.25855e6i	0.541394i
$353$	2.62106e6i	1.11954i	0.828648	+	0.559771i	$0.189111\pi$
$353$	2.62106e6i	1.11954i	−0.828648	+	0.559771i	$0.810889\pi$
$354$	−1.11335e6	−0.472195
$355$	0	0
$356$	−2.77570e6	−1.16077
$357$	− 1.77490e6i	− 0.737059i
$358$	− 4.64364e6i	− 1.91492i
$359$	818855.	0.335329	0.167664	−	0.985844i	$-0.446377\pi$
$359$	818855.	0.335329	0.167664	+	0.985844i	$0.446377\pi$
$360$	0	0
$361$	1.49300e6	0.602966
$362$	− 1.05828e6i	− 0.424451i
$363$	1.86592e6i	0.743235i
$364$	−480957.	−0.190262
$365$	0	0
$366$	2.99575e6	1.16897
$367$	2.81469e6i	1.09085i	0.838160	+	0.545425i	$0.183632\pi$
$367$	2.81469e6i	1.09085i	−0.838160	+	0.545425i	$0.816368\pi$
$368$	2.62472e6i	1.01033i
$369$	−191263.	−0.0731249
$370$	0	0
$371$	−3.24730e6	−1.22486
$372$	− 2.97839e6i	− 1.11590i
$373$	− 4.64910e6i	− 1.73020i	−0.501598	−	0.865101i	$-0.667254\pi$
$373$	− 4.64910e6i	− 1.73020i	0.501598	−	0.865101i	$-0.332746\pi$
$374$	1.21802e6	0.450271
$375$	0	0
$376$	1.36200e6	0.496829
$377$	7835.20i	0.00283920i
$378$	3.86081e6i	1.38979i
$379$	2.33478e6	0.834924	0.417462	−	0.908694i	$-0.362920\pi$
$379$	2.33478e6	0.834924	0.417462	+	0.908694i	$0.362920\pi$
$380$	0	0
$381$	−4.51482e6	−1.59341
$382$	7.04394e6i	2.46977i
$383$	− 3.05707e6i	− 1.06490i	−0.846462	−	0.532449i	$-0.821272\pi$
$383$	− 3.05707e6i	− 1.06490i	0.846462	−	0.532449i	$-0.178728\pi$
$384$	2.31053e6	0.799621
$385$	0	0
$386$	−66291.6	−0.0226459
$387$	73806.0i	0.0250504i
$388$	407083.i	0.137279i
$389$	−1.25272e6	−0.419741	−0.209871	−	0.977729i	$-0.567304\pi$
$389$	−1.25272e6	−0.419741	−0.209871	+	0.977729i	$0.567304\pi$
$390$	0	0
$391$	1.82889e6	0.604986
$392$	121833.i	0.0400450i
$393$	1.50303e6i	0.490891i
$394$	4.13018e6	1.34038
$395$	0	0
$396$	−73172.4	−0.0234482
$397$	− 5.44444e6i	− 1.73371i	−0.498559	−	0.866856i	$-0.666137\pi$
$397$	− 5.44444e6i	− 1.73371i	0.498559	−	0.866856i	$-0.333863\pi$
$398$	− 2.33952e6i	− 0.740318i
$399$	4.04427e6	1.27177
$400$	0	0
$401$	−3.75387e6	−1.16578	−0.582892	−	0.812549i	$-0.698079\pi$
$401$	−3.75387e6	−1.16578	−0.582892	+	0.812549i	$0.698079\pi$
$402$	− 34701.2i	− 0.0107098i
$403$	− 1.59712e6i	− 0.489864i
$404$	1.90001e6	0.579167
$405$	0	0
$406$	−45706.0	−0.0137613
$407$	645663.i	0.193206i
$408$	− 1.04812e6i	− 0.311718i
$409$	−4.09010e6	−1.20900	−0.604500	−	0.796605i	$-0.706627\pi$
$409$	−4.09010e6	−1.20900	−0.604500	+	0.796605i	$0.706627\pi$
$410$	0	0
$411$	−1.08518e6	−0.316882
$412$	− 1.25619e6i	− 0.364596i
$413$	− 1.38070e6i	− 0.398313i
$414$	−277135.	−0.0794677
$415$	0	0
$416$	−1.11174e6	−0.314970
$417$	4.45094e6i	1.25346i
$418$	2.77536e6i	0.776925i
$419$	−3.67162e6	−1.02170	−0.510849	−	0.859671i	$-0.670669\pi$
$419$	−3.67162e6	−1.02170	−0.510849	+	0.859671i	$0.670669\pi$
$420$	0	0
$421$	1.59907e6	0.439705	0.219852	−	0.975533i	$-0.429442\pi$
$421$	1.59907e6	0.439705	0.219852	+	0.975533i	$0.429442\pi$
$422$	1.20493e6i	0.329367i
$423$	309964.i	0.0842288i
$424$	−1.91761e6	−0.518020
$425$	0	0
$426$	−4.33037e6	−1.15612
$427$	3.71514e6i	0.986065i
$428$	3.75009e6i	0.989536i
$429$	484779.	0.127175
$430$	0	0
$431$	6.53931e6	1.69566	0.847830	−	0.530268i	$-0.177909\pi$
$431$	6.53931e6	1.69566	0.847830	+	0.530268i	$0.177909\pi$
$432$	4.91409e6i	1.26688i
$433$	− 1.60206e6i	− 0.410639i	−0.978695	−	0.205319i	$-0.934177\pi$
$433$	− 1.60206e6i	− 0.410639i	0.978695	−	0.205319i	$-0.0658233\pi$
$434$	9.31668e6	2.37431
$435$	0	0
$436$	−792226.	−0.199587
$437$	4.16729e6i	1.04388i
$438$	− 520114.i	− 0.129543i
$439$	4.84652e6	1.20024	0.600121	−	0.799909i	$-0.295119\pi$
$439$	4.84652e6	1.20024	0.600121	+	0.799909i	$0.295119\pi$
$440$	0	0
$441$	−27726.7	−0.00678894
$442$	1.07593e6i	0.261957i
$443$	− 580247.i	− 0.140476i	−0.997530	−	0.0702382i	$-0.977624\pi$
$443$	− 580247.i	− 0.140476i	0.997530	−	0.0702382i	$-0.0223759\pi$
$444$	−1.06361e6	−0.256050
$445$	0	0
$446$	−8.30997e6	−1.97816
$447$	243784.i	0.0577081i
$448$	− 1.04878e6i	− 0.246881i
$449$	−2.04347e6	−0.478356	−0.239178	−	0.970976i	$-0.576878\pi$
$449$	−2.04347e6	−0.478356	−0.239178	+	0.970976i	$0.576878\pi$
$450$	0	0
$451$	−2.01104e6	−0.465565
$452$	811711.i	0.186877i
$453$	− 5.74993e6i	− 1.31649i
$454$	−4.17640e6	−0.950960
$455$	0	0
$456$	2.38824e6	0.537857
$457$	1.52332e6i	0.341194i	0.985341	+	0.170597i	$0.0545695\pi$
$457$	1.52332e6i	0.341194i	−0.985341	+	0.170597i	$0.945430\pi$
$458$	8.12951e6i	1.81093i
$459$	3.42411e6	0.758605
$460$	0	0
$461$	7.87768e6	1.72642	0.863209	−	0.504847i	$-0.168451\pi$
$461$	7.87768e6	1.72642	0.863209	+	0.504847i	$0.168451\pi$
$462$	2.82792e6i	0.616399i
$463$	− 136322.i	− 0.0295539i	−0.999891	−	0.0147770i	$-0.995296\pi$
$463$	− 136322.i	− 0.0295539i	0.999891	−	0.0147770i	$-0.00470382\pi$
$464$	−58175.3	−0.0125442
$465$	0	0
$466$	−7.90420e6	−1.68614
$467$	5.56784e6i	1.18139i	0.806894	+	0.590697i	$0.201147\pi$
$467$	5.56784e6i	1.18139i	−0.806894	+	0.590697i	$0.798853\pi$
$468$	− 64636.8i	− 0.0136416i
$469$	43034.2	0.00903404
$470$	0	0
$471$	−3.53100e6	−0.733407
$472$	− 815339.i	− 0.168455i
$473$	776035.i	0.159488i
$474$	−8.13852e6	−1.66379
$475$	0	0
$476$	−2.48828e6	−0.503363
$477$	− 436411.i	− 0.0878213i
$478$	3.62284e6i	0.725237i
$479$	−3.92083e6	−0.780799	−0.390399	−	0.920646i	$-0.627663\pi$
$479$	−3.92083e6	−0.780799	−0.390399	+	0.920646i	$0.627663\pi$
$480$	0	0
$481$	−570345.	−0.112402
$482$	1.22970e7i	2.41091i
$483$	4.24621e6i	0.828197i
$484$	2.61588e6	0.507581
$485$	0	0
$486$	−1.00158e6	−0.192351
$487$	8.77882e6i	1.67731i	0.544661	+	0.838656i	$0.316658\pi$
$487$	8.77882e6i	1.67731i	−0.544661	+	0.838656i	$0.683342\pi$
$488$	2.19389e6i	0.417027i
$489$	3.07614e6	0.581747
$490$	0	0
$491$	−9.13377e6	−1.70980	−0.854902	−	0.518789i	$-0.826383\pi$
$491$	−9.13377e6	−1.70980	−0.854902	+	0.518789i	$0.826383\pi$
$492$	− 3.31282e6i	− 0.616999i
$493$	40536.2i	0.00751148i
$494$	−2.45161e6	−0.451996
$495$	0	0
$496$	1.18584e7	2.16432
$497$	− 5.37026e6i	− 0.975223i
$498$	1.55405e6i	0.280796i
$499$	9.41419e6	1.69251	0.846256	−	0.532777i	$-0.178852\pi$
$499$	9.41419e6	1.69251	0.846256	+	0.532777i	$0.178852\pi$
$500$	0	0
$501$	−1.03403e6	−0.184052
$502$	2.61180e6i	0.462574i
$503$	− 5.68086e6i	− 1.00114i	−0.865697	−	0.500569i	$-0.833124\pi$
$503$	− 5.68086e6i	− 1.00114i	0.865697	−	0.500569i	$-0.166876\pi$
$504$	−196963.	−0.0345389
$505$	0	0
$506$	−2.91395e6	−0.505947
$507$	428229.i	0.0739871i
$508$	6.32945e6i	1.08819i
$509$	−2.85772e6	−0.488906	−0.244453	−	0.969661i	$-0.578608\pi$
$509$	−2.85772e6	−0.488906	−0.244453	+	0.969661i	$0.578608\pi$
$510$	0	0
$511$	645012.	0.109274
$512$	− 5.04413e6i	− 0.850377i
$513$	7.80215e6i	1.30894i
$514$	−5.21195e6	−0.870147
$515$	0	0
$516$	−1.27837e6	−0.211365
$517$	3.25913e6i	0.536260i
$518$	− 3.32706e6i	− 0.544799i
$519$	9.19185e6	1.49791
$520$	0	0
$521$	2.64216e6	0.426446	0.213223	−	0.977004i	$-0.431604\pi$
$521$	2.64216e6	0.426446	0.213223	+	0.977004i	$0.431604\pi$
$522$	− 6142.52i	0 0.000986667i
$523$	− 1.02124e7i	− 1.63258i	−0.577645	−	0.816288i	$-0.696028\pi$
$523$	− 1.02124e7i	− 1.63258i	0.577645	−	0.816288i	$-0.303972\pi$
$524$	2.10714e6	0.335247
$525$	0	0
$526$	1.22320e6	0.192767
$527$	− 8.26286e6i	− 1.29600i
$528$	3.59942e6i	0.561885i
$529$	2.06096e6	0.320207
$530$	0	0
$531$	185555.	0.0285586
$532$	− 5.66977e6i	− 0.868533i
$533$	− 1.77645e6i	− 0.270854i
$534$	−1.44167e7	−2.18782
$535$	0	0
$536$	25412.8	0.00382068
$537$	− 9.56186e6i	− 1.43089i
$538$	1.40652e7i	2.09504i
$539$	−291533.	−0.0432232
$540$	0	0
$541$	−2.02275e6	−0.297132	−0.148566	−	0.988902i	$-0.547466\pi$
$541$	−2.02275e6	−0.297132	−0.148566	+	0.988902i	$0.547466\pi$
$542$	− 1.48686e7i	− 2.17407i
$543$	− 2.17913e6i	− 0.317163i
$544$	−5.75169e6	−0.833293
$545$	0	0
$546$	−2.49804e6	−0.358606
$547$	2.12905e6i	0.304241i	0.988362	+	0.152120i	$0.0486101\pi$
$547$	2.12905e6i	0.304241i	−0.988362	+	0.152120i	$0.951390\pi$
$548$	1.52135e6i	0.216410i
$549$	−499285.	−0.0706998
$550$	0	0
$551$	−92365.4	−0.0129608
$552$	2.50750e6i	0.350262i
$553$	− 1.00929e7i	− 1.40347i
$554$	−7.71242e6	−1.06762
$555$	0	0
$556$	6.23989e6	0.856032
$557$	− 8.58830e6i	− 1.17292i	−0.809977	−	0.586461i	$-0.800521\pi$
$557$	− 8.58830e6i	− 1.17292i	0.809977	−	0.586461i	$-0.199479\pi$
$558$	1.25209e6i	0.170235i
$559$	−685510.	−0.0927863
$560$	0	0
$561$	2.50805e6	0.336457
$562$	6.81238e6i	0.909826i
$563$	− 1.02986e7i	− 1.36933i	−0.728859	−	0.684664i	$-0.759949\pi$
$563$	− 1.02986e7i	− 1.36933i	0.728859	−	0.684664i	$-0.240051\pi$
$564$	−5.36880e6	−0.710689
$565$	0	0
$566$	−6.02361e6	−0.790343
$567$	7.35126e6i	0.960294i
$568$	− 3.17127e6i	− 0.412442i
$569$	−4.16791e6	−0.539681	−0.269841	−	0.962905i	$-0.586971\pi$
$569$	−4.16791e6	−0.539681	−0.269841	+	0.962905i	$0.586971\pi$
$570$	0	0
$571$	7.69686e6	0.987923	0.493962	−	0.869484i	$-0.335548\pi$
$571$	7.69686e6	0.987923	0.493962	+	0.869484i	$0.335548\pi$
$572$	− 679625.i	− 0.0868519i
$573$	1.45044e7i	1.84549i
$574$	1.03628e7	1.31280
$575$	0	0
$576$	140947.	0.0177011
$577$	− 1.42934e7i	− 1.78729i	−0.448775	−	0.893645i	$-0.648140\pi$
$577$	− 1.42934e7i	− 1.78729i	0.448775	−	0.893645i	$-0.351860\pi$
$578$	− 4.77220e6i	− 0.594154i
$579$	−136503.	−0.0169217
$580$	0	0
$581$	−1.92723e6	−0.236861
$582$	2.11435e6i	0.258743i
$583$	− 4.58866e6i	− 0.559133i
$584$	380897.	0.0462141
$585$	0	0
$586$	1.29264e7	1.55501
$587$	− 1.52860e7i	− 1.83104i	−0.402268	−	0.915522i	$-0.631778\pi$
$587$	− 1.52860e7i	− 1.83104i	0.402268	−	0.915522i	$-0.368222\pi$
$588$	− 480247.i	− 0.0572824i
$589$	1.88277e7	2.23619
$590$	0	0
$591$	8.50457e6	1.00157
$592$	− 4.23473e6i	− 0.496617i
$593$	8.42604e6i	0.983981i	0.870601	+	0.491991i	$0.163731\pi$
$593$	8.42604e6i	0.983981i	−0.870601	+	0.491991i	$0.836269\pi$
$594$	−5.45559e6	−0.634418
$595$	0	0
$596$	341768.	0.0394109
$597$	− 4.81737e6i	− 0.553189i
$598$	− 2.57403e6i	− 0.294348i
$599$	1.40447e6	0.159936	0.0799680	−	0.996797i	$-0.474518\pi$
$599$	1.40447e6	0.159936	0.0799680	+	0.996797i	$0.474518\pi$
$600$	0	0
$601$	3.45723e6	0.390429	0.195214	−	0.980761i	$-0.437460\pi$
$601$	3.45723e6	0.390429	0.195214	+	0.980761i	$0.437460\pi$
$602$	− 3.99887e6i	− 0.449723i
$603$	5783.46i	0 0.000647731i
$604$	−8.06098e6	−0.899075
$605$	0	0
$606$	9.86848e6	1.09161
$607$	− 1.56253e7i	− 1.72130i	−0.509194	−	0.860652i	$-0.670056\pi$
$607$	− 1.56253e7i	− 1.72130i	0.509194	−	0.860652i	$-0.329944\pi$
$608$	− 1.31058e7i	− 1.43782i
$609$	−94114.5	−0.0102828
$610$	0	0
$611$	−2.87895e6	−0.311983
$612$	− 334405.i	− 0.0360906i
$613$	3.97422e6i	0.427170i	0.976924	+	0.213585i	$0.0685141\pi$
$613$	3.97422e6i	0.427170i	−0.976924	+	0.213585i	$0.931486\pi$
$614$	2.13494e6	0.228541
$615$	0	0
$616$	−2.07098e6	−0.219899
$617$	− 5.17583e6i	− 0.547353i	−0.961822	−	0.273676i	$-0.911760\pi$
$617$	− 5.17583e6i	− 0.547353i	0.961822	−	0.273676i	$-0.0882397\pi$
$618$	− 6.52451e6i	− 0.687189i
$619$	−2.06480e6	−0.216597	−0.108298	−	0.994118i	$-0.534540\pi$
$619$	−2.06480e6	−0.216597	−0.108298	+	0.994118i	$0.534540\pi$
$620$	0	0
$621$	−8.19174e6	−0.852407
$622$	1.70552e7i	1.76759i
$623$	− 1.78787e7i	− 1.84550i
$624$	−3.17954e6	−0.326891
$625$	0	0
$626$	8.78390e6	0.895883
$627$	5.71483e6i	0.580543i
$628$	4.95020e6i	0.500869i
$629$	−2.95074e6	−0.297374
$630$	0	0
$631$	−1.72030e7	−1.72001	−0.860003	−	0.510289i	$-0.829539\pi$
$631$	−1.72030e7	−1.72001	−0.860003	+	0.510289i	$0.829539\pi$
$632$	− 5.96010e6i	− 0.593555i
$633$	2.48110e6i	0.246113i
$634$	1.61626e7	1.59693
$635$	0	0
$636$	7.55896e6	0.741002
$637$	− 257526.i	− 0.0251462i
$638$	− 64585.8i	− 0.00628182i
$639$	721720.	0.0699224
$640$	0	0
$641$	−7.98432e6	−0.767526	−0.383763	−	0.923432i	$-0.625372\pi$
$641$	−7.98432e6	−0.767526	−0.383763	+	0.923432i	$0.625372\pi$
$642$	1.94775e7i	1.86508i
$643$	5.66314e6i	0.540169i	0.962837	+	0.270085i	$0.0870517\pi$
$643$	5.66314e6i	0.540169i	−0.962837	+	0.270085i	$0.912948\pi$
$644$	5.95288e6	0.565604
$645$	0	0
$646$	−1.26837e7	−1.19581
$647$	− 3.74193e6i	− 0.351427i	−0.984441	−	0.175713i	$-0.943777\pi$
$647$	− 3.74193e6i	− 0.351427i	0.984441	−	0.175713i	$-0.0562232\pi$
$648$	4.34111e6i	0.406128i
$649$	1.95103e6	0.181824
$650$	0	0
$651$	1.91842e7	1.77416
$652$	− 4.31253e6i	− 0.397295i
$653$	− 1.41967e7i	− 1.30288i	−0.758702	−	0.651438i	$-0.774166\pi$
$653$	− 1.41967e7i	− 1.30288i	0.758702	−	0.651438i	$-0.225834\pi$
$654$	−4.11474e6	−0.376182
$655$	0	0
$656$	1.31899e7	1.19669
$657$	86684.6i	0.00783481i
$658$	− 1.67941e7i	− 1.51214i
$659$	7.43083e6	0.666536	0.333268	−	0.942832i	$-0.391849\pi$
$659$	7.43083e6	0.666536	0.333268	+	0.942832i	$0.391849\pi$
$660$	0	0
$661$	1.44503e7	1.28639	0.643194	−	0.765703i	$-0.277609\pi$
$661$	1.44503e7	1.28639	0.643194	+	0.765703i	$0.277609\pi$
$662$	− 1.00015e7i	− 0.886989i
$663$	2.21548e6i	0.195742i
$664$	−1.13808e6	−0.100174
$665$	0	0
$666$	447131.	0.0390615
$667$	− 96977.5i	− 0.00844027i
$668$	1.44964e6i	0.125695i
$669$	−1.71113e7	−1.47815
$670$	0	0
$671$	−5.24975e6	−0.450125
$672$	− 1.33539e7i	− 1.14074i
$673$	8.38558e6i	0.713667i	0.934168	+	0.356834i	$0.116144\pi$
$673$	8.38558e6i	0.713667i	−0.934168	+	0.356834i	$0.883856\pi$
$674$	−3.12143e6	−0.264669
$675$	0	0
$676$	600346.	0.0505283
$677$	1.53211e7i	1.28475i	0.766390	+	0.642376i	$0.222051\pi$
$677$	1.53211e7i	1.28475i	−0.766390	+	0.642376i	$0.777949\pi$
$678$	4.21594e6i	0.352225i
$679$	−2.62208e6	−0.218259
$680$	0	0
$681$	−8.59974e6	−0.710587
$682$	1.31651e7i	1.08384i
$683$	− 1.08140e7i	− 0.887021i	−0.896269	−	0.443510i	$-0.853733\pi$
$683$	− 1.08140e7i	− 0.887021i	0.896269	−	0.443510i	$-0.146267\pi$
$684$	761972.	0.0622729
$685$	0	0
$686$	−1.50669e7	−1.22240
$687$	1.67397e7i	1.35318i
$688$	− 5.08981e6i	− 0.409950i
$689$	4.05339e6	0.325290
$690$	0	0
$691$	2.25890e7	1.79971	0.899853	−	0.436193i	$-0.143673\pi$
$691$	2.25890e7	1.79971	0.899853	+	0.436193i	$0.143673\pi$
$692$	− 1.28863e7i	− 1.02297i
$693$	− 471314.i	− 0.0372801i
$694$	−2.43237e7	−1.91704
$695$	0	0
$696$	−55577.1	−0.00434883
$697$	− 9.19065e6i	− 0.716579i
$698$	− 1.20758e7i	− 0.938159i
$699$	−1.62758e7	−1.25994
$700$	0	0
$701$	1.06616e7	0.819461	0.409731	−	0.912207i	$-0.365623\pi$
$701$	1.06616e7	0.819461	0.409731	+	0.912207i	$0.365623\pi$
$702$	− 4.81919e6i	− 0.369089i
$703$	− 6.72353e6i	− 0.513108i
$704$	1.48200e6	0.112698
$705$	0	0
$706$	1.90852e7	1.44107
$707$	1.22383e7i	0.920812i
$708$	3.21395e6i	0.240966i
$709$	1.17029e7	0.874333	0.437166	−	0.899381i	$-0.355982\pi$
$709$	1.17029e7	0.874333	0.437166	+	0.899381i	$0.355982\pi$
$710$	0	0
$711$	1.35640e6	0.100627
$712$	− 1.05578e7i	− 0.780501i
$713$	1.97678e7i	1.45625i
$714$	−1.29238e7	−0.948738
$715$	0	0
$716$	−1.34050e7	−0.977204
$717$	7.45990e6i	0.541920i
$718$	− 5.96247e6i	− 0.431633i
$719$	−2.34149e6	−0.168915	−0.0844577	−	0.996427i	$-0.526916\pi$
$719$	−2.34149e6	−0.168915	−0.0844577	+	0.996427i	$0.526916\pi$
$720$	0	0
$721$	8.09128e6	0.579668
$722$	− 1.08713e7i	− 0.776135i
$723$	2.53211e7i	1.80151i
$724$	−3.05498e6	−0.216602
$725$	0	0
$726$	1.35866e7	0.956687
$727$	− 6.10487e6i	− 0.428391i	−0.976791	−	0.214195i	$-0.931287\pi$
$727$	− 6.10487e6i	− 0.428391i	0.976791	−	0.214195i	$-0.0687129\pi$
$728$	− 1.82940e6i	− 0.127932i
$729$	−1.52564e7	−1.06325
$730$	0	0
$731$	−3.54655e6	−0.245478
$732$	− 8.64799e6i	− 0.596537i
$733$	− 1.79993e7i	− 1.23736i	−0.785642	−	0.618681i	$-0.787667\pi$
$733$	− 1.79993e7i	− 1.23736i	0.785642	−	0.618681i	$-0.212333\pi$
$734$	2.04951e7	1.40414
$735$	0	0
$736$	1.37602e7	0.936330
$737$	60810.4i	0.00412391i
$738$	1.39268e6i	0.0941260i
$739$	−2.02603e7	−1.36469	−0.682346	−	0.731030i	$-0.739040\pi$
$739$	−2.02603e7	−1.36469	−0.682346	+	0.731030i	$0.739040\pi$
$740$	0	0
$741$	−5.04819e6	−0.337746
$742$	2.36451e7i	1.57664i
$743$	2.77445e7i	1.84376i	0.387472	+	0.921881i	$0.373348\pi$
$743$	2.77445e7i	1.84376i	−0.387472	+	0.921881i	$0.626652\pi$
$744$	1.13288e7	0.750328
$745$	0	0
$746$	−3.38523e7	−2.22711
$747$	− 259005.i	− 0.0169827i
$748$	− 3.51611e6i	− 0.229778i
$749$	−2.41548e7	−1.57326
$750$	0	0
$751$	−2.16593e7	−1.40134	−0.700672	−	0.713483i	$-0.747117\pi$
$751$	−2.16593e7	−1.40134	−0.700672	+	0.713483i	$0.747117\pi$
$752$	− 2.13758e7i	− 1.37841i
$753$	5.37804e6i	0.345650i
$754$	57051.7	0.00365461
$755$	0	0
$756$	1.11452e7	0.709223
$757$	− 2.34452e7i	− 1.48701i	−0.668731	−	0.743505i	$-0.733162\pi$
$757$	− 2.34452e7i	− 1.48701i	0.668731	−	0.743505i	$-0.266838\pi$
$758$	− 1.70006e7i	− 1.07471i
$759$	−6.00019e6	−0.378060
$760$	0	0
$761$	−2.00518e7	−1.25514	−0.627569	−	0.778561i	$-0.715950\pi$
$761$	−2.00518e7	−1.25514	−0.627569	+	0.778561i	$0.715950\pi$
$762$	3.28745e7i	2.05103i
$763$	− 5.10284e6i	− 0.317322i
$764$	2.03341e7	1.26035
$765$	0	0
$766$	−2.22599e7	−1.37073
$767$	1.72344e6i	0.105781i
$768$	− 2.05407e7i	− 1.25664i
$769$	−2.69938e7	−1.64607	−0.823034	−	0.567992i	$-0.807720\pi$
$769$	−2.69938e7	−1.64607	−0.823034	+	0.567992i	$0.807720\pi$
$770$	0	0
$771$	−1.07321e7	−0.650201
$772$	191367.i	0.0115564i
$773$	1.55609e7i	0.936668i	0.883552	+	0.468334i	$0.155146\pi$
$773$	1.55609e7i	0.936668i	−0.883552	+	0.468334i	$0.844854\pi$
$774$	537416.	0.0322447
$775$	0	0
$776$	−1.54841e6	−0.0923061
$777$	− 6.85085e6i	− 0.407091i
$778$	9.12167e6i	0.540288i
$779$	2.09418e7	1.23643
$780$	0	0
$781$	7.58855e6	0.445175
$782$	− 1.33170e7i	− 0.778734i
$783$	− 181565.i	− 0.0105834i
$784$	1.91209e6	0.111101
$785$	0	0
$786$	1.09442e7	0.631872
$787$	2.60132e7i	1.49712i	0.663067	+	0.748560i	$0.269255\pi$
$787$	2.60132e7i	1.49712i	−0.663067	+	0.748560i	$0.730745\pi$
$788$	− 1.19228e7i	− 0.684010i
$789$	2.51872e6	0.144041
$790$	0	0
$791$	−5.22834e6	−0.297114
$792$	− 278323.i	− 0.0157665i
$793$	− 4.63736e6i	− 0.261871i
$794$	−3.96435e7	−2.23162
$795$	0	0
$796$	−6.75360e6	−0.377792
$797$	1.36342e7i	0.760296i	0.924926	+	0.380148i	$0.124127\pi$
$797$	1.36342e7i	0.760296i	−0.924926	+	0.380148i	$0.875873\pi$
$798$	− 2.94482e7i	− 1.63701i
$799$	−1.48945e7	−0.825390
$800$	0	0
$801$	2.40275e6	0.132320
$802$	2.73337e7i	1.50059i
$803$	911448.i	0.0498819i
$804$	−100174.	−0.00546530
$805$	0	0
$806$	−1.16294e7	−0.630550
$807$	2.89621e7i	1.56548i
$808$	7.22701e6i	0.389431i
$809$	7.65343e6	0.411135	0.205568	−	0.978643i	$-0.434096\pi$
$809$	7.65343e6	0.411135	0.205568	+	0.978643i	$0.434096\pi$
$810$	0	0
$811$	1.95683e7	1.04472	0.522360	−	0.852725i	$-0.325052\pi$
$811$	1.95683e7	1.04472	0.522360	+	0.852725i	$0.325052\pi$
$812$	131942.i	0.00702251i
$813$	− 3.06164e7i	− 1.62453i
$814$	4.70137e6	0.248693
$815$	0	0
$816$	−1.64497e7	−0.864832
$817$	− 8.08115e6i	− 0.423563i
$818$	2.97820e7i	1.55622i
$819$	416334.	0.0216886
$820$	0	0
$821$	−2.71975e7	−1.40822	−0.704112	−	0.710089i	$-0.748654\pi$
$821$	−2.71975e7	−1.40822	−0.704112	+	0.710089i	$0.748654\pi$
$822$	7.90172e6i	0.407889i
$823$	3.39792e7i	1.74870i	0.485300	+	0.874348i	$0.338710\pi$
$823$	3.39792e7i	1.74870i	−0.485300	+	0.874348i	$0.661290\pi$
$824$	4.77811e6	0.245153
$825$	0	0
$826$	−1.00535e7	−0.512706
$827$	2.56121e7i	1.30221i	0.758988	+	0.651105i	$0.225694\pi$
$827$	2.56121e7i	1.30221i	−0.758988	+	0.651105i	$0.774306\pi$
$828$	800020.i	0.0405532i
$829$	−1.33798e7	−0.676180	−0.338090	−	0.941114i	$-0.609781\pi$
$829$	−1.33798e7	−0.676180	−0.338090	+	0.941114i	$0.609781\pi$
$830$	0	0
$831$	−1.58809e7	−0.797759
$832$	1.30912e6i	0.0655648i
$833$	− 1.33233e6i	− 0.0665274i
$834$	3.24094e7	1.61345
$835$	0	0
$836$	8.01178e6	0.396473
$837$	3.70100e7i	1.82602i
$838$	2.67348e7i	1.31512i
$839$	1.59865e7	0.784056	0.392028	−	0.919953i	$-0.371774\pi$
$839$	1.59865e7	0.784056	0.392028	+	0.919953i	$0.371774\pi$
$840$	0	0
$841$	−2.05090e7	−0.999895
$842$	− 1.16435e7i	− 0.565985i
$843$	1.40276e7i	0.679851i
$844$	3.47832e6	0.168079
$845$	0	0
$846$	2.25699e6	0.108419
$847$	1.68493e7i	0.806998i
$848$	3.00958e7i	1.43720i
$849$	−1.24034e7	−0.590570
$850$	0	0
$851$	7.05925e6	0.334145
$852$	1.25007e7i	0.589978i
$853$	2.04688e7i	0.963209i	0.876389	+	0.481605i	$0.159946\pi$
$853$	2.04688e7i	0.963209i	−0.876389	+	0.481605i	$0.840054\pi$
$854$	2.70517e7	1.26926
$855$	0	0
$856$	−1.42640e7	−0.665362
$857$	− 1.31659e7i	− 0.612349i	−0.951975	−	0.306175i	$-0.900951\pi$
$857$	− 1.31659e7i	− 0.612349i	0.951975	−	0.306175i	$-0.0990492\pi$
$858$	− 3.52990e6i	− 0.163698i
$859$	3.58429e7	1.65737	0.828686	−	0.559713i	$-0.189089\pi$
$859$	3.58429e7	1.65737	0.828686	+	0.559713i	$0.189089\pi$
$860$	0	0
$861$	2.13383e7	0.980962
$862$	− 4.76158e7i	− 2.18264i
$863$	9.29285e6i	0.424739i	0.977189	+	0.212369i	$0.0681180\pi$
$863$	9.29285e6i	0.424739i	−0.977189	+	0.212369i	$0.931882\pi$
$864$	2.57623e7	1.17409
$865$	0	0
$866$	−1.16654e7	−0.528572
$867$	− 9.82658e6i	− 0.443971i
$868$	− 2.68949e7i	− 1.21163i
$869$	1.42619e7	0.640662
$870$	0	0
$871$	−53716.8	−0.00239919
$872$	− 3.01336e6i	− 0.134202i
$873$	− 352387.i	− 0.0156489i
$874$	3.03440e7	1.34368
$875$	0	0
$876$	−1.50144e6	−0.0661070
$877$	3.01018e7i	1.32158i	0.750570	+	0.660791i	$0.229779\pi$
$877$	3.01018e7i	1.32158i	−0.750570	+	0.660791i	$0.770221\pi$
$878$	− 3.52898e7i	− 1.54494i
$879$	2.66171e7	1.16195
$880$	0	0
$881$	4.75023e6	0.206193	0.103097	−	0.994671i	$-0.467125\pi$
$881$	4.75023e6	0.206193	0.103097	+	0.994671i	$0.467125\pi$
$882$	201891.i	0.00873868i
$883$	2.31803e7i	1.00050i	0.865881	+	0.500250i	$0.166758\pi$
$883$	2.31803e7i	1.00050i	−0.865881	+	0.500250i	$0.833242\pi$
$884$	3.10595e6	0.133679
$885$	0	0
$886$	−4.22505e6	−0.180820
$887$	− 3.60486e7i	− 1.53844i	−0.638986	−	0.769219i	$-0.720646\pi$
$887$	− 3.60486e7i	− 1.53844i	0.638986	−	0.769219i	$-0.279354\pi$
$888$	− 4.04560e6i	− 0.172167i
$889$	−4.07689e7	−1.73011
$890$	0	0
$891$	−1.03879e7	−0.438361
$892$	2.39888e7i	1.00948i
$893$	− 3.39385e7i	− 1.42418i
$894$	1.77511e6	0.0742816
$895$	0	0
$896$	2.08641e7	0.868221
$897$	− 5.30026e6i	− 0.219946i
$898$	1.48794e7i	0.615737i
$899$	−438142.	−0.0180807
$900$	0	0
$901$	2.09706e7	0.860595
$902$	1.46433e7i	0.599272i
$903$	− 8.23418e6i	− 0.336048i
$904$	−3.08747e6	−0.125656
$905$	0	0
$906$	−4.18679e7	−1.69457
$907$	− 3.74734e7i	− 1.51253i	−0.654264	−	0.756266i	$-0.727021\pi$
$907$	− 3.74734e7i	− 1.51253i	0.654264	−	0.756266i	$-0.272979\pi$
$908$	1.20562e7i	0.485284i
$909$	−1.64473e6	−0.0660212
$910$	0	0
$911$	2.58798e7	1.03315	0.516577	−	0.856241i	$-0.327206\pi$
$911$	2.58798e7	1.03315	0.516577	+	0.856241i	$0.327206\pi$
$912$	− 3.74821e7i	− 1.49223i
$913$	− 2.72332e6i	− 0.108124i
$914$	1.10920e7	0.439182
$915$	0	0
$916$	2.34679e7	0.924134
$917$	1.35724e7i	0.533006i
$918$	− 2.49325e7i	− 0.976472i
$919$	−1.10504e6	−0.0431606	−0.0215803	−	0.999767i	$-0.506870\pi$
$919$	−1.10504e6	−0.0431606	−0.0215803	+	0.999767i	$0.506870\pi$
$920$	0	0
$921$	4.39611e6	0.170773
$922$	− 5.73611e7i	− 2.22224i
$923$	6.70333e6i	0.258992i
$924$	8.16350e6	0.314555
$925$	0	0
$926$	−992628.	−0.0380416
$927$	1.08740e6i	0.0415615i
$928$	304985.i	0.0116254i
$929$	2.92574e6	0.111224	0.0556118	−	0.998452i	$-0.482289\pi$
$929$	2.92574e6	0.111224	0.0556118	+	0.998452i	$0.482289\pi$
$930$	0	0
$931$	3.03585e6	0.114790
$932$	2.28175e7i	0.860453i
$933$	3.51189e7i	1.32080i
$934$	4.05421e7	1.52068
$935$	0	0
$936$	245856.	0.00917258
$937$	2.45270e7i	0.912632i	0.889818	+	0.456316i	$0.150831\pi$
$937$	2.45270e7i	0.912632i	−0.889818	+	0.456316i	$0.849169\pi$
$938$	− 313352.i	− 0.0116286i
$939$	1.80872e7	0.669432
$940$	0	0
$941$	919876.	0.0338653	0.0169327	−	0.999857i	$-0.494610\pi$
$941$	919876.	0.0338653	0.0169327	+	0.999857i	$0.494610\pi$
$942$	2.57108e7i	0.944037i
$943$	2.19874e7i	0.805184i
$944$	−1.27963e7	−0.467362
$945$	0	0
$946$	5.65068e6	0.205292
$947$	9.61521e6i	0.348405i	0.984710	+	0.174202i	$0.0557347\pi$
$947$	9.61521e6i	0.348405i	−0.984710	+	0.174202i	$0.944265\pi$
$948$	2.34939e7i	0.849051i
$949$	−805126.	−0.0290201
$950$	0	0
$951$	3.32808e7	1.19328
$952$	− 9.46455e6i	− 0.338460i
$953$	− 2.43193e7i	− 0.867399i	−0.901058	−	0.433699i	$-0.857208\pi$
$953$	− 2.43193e7i	− 0.867399i	0.901058	−	0.433699i	$-0.142792\pi$
$954$	−3.17772e6	−0.113043
$955$	0	0
$956$	1.04582e7	0.370096
$957$	− 132990.i	− 0.00469397i
$958$	2.85494e7i	1.00504i
$959$	−9.79921e6	−0.344068
$960$	0	0
$961$	6.06813e7	2.11956
$962$	4.15295e6i	0.144683i
$963$	− 3.24622e6i	− 0.112801i
$964$	3.54984e7	1.23031
$965$	0	0
$966$	3.09186e7	1.06605
$967$	− 2.12248e7i	− 0.729922i	−0.931023	−	0.364961i	$-0.881082\pi$
$967$	− 2.12248e7i	− 0.729922i	0.931023	−	0.364961i	$-0.118918\pi$
$968$	9.94993e6i	0.341296i
$969$	−2.61173e7	−0.893549
$970$	0	0
$971$	−3.64590e7	−1.24096	−0.620479	−	0.784223i	$-0.713062\pi$
$971$	−3.64590e7	−1.24096	−0.620479	+	0.784223i	$0.713062\pi$
$972$	2.89131e6i	0.0981587i
$973$	4.01920e7i	1.36100i
$974$	6.39227e7	2.15903
$975$	0	0
$976$	3.44318e7	1.15700
$977$	− 1.86618e7i	− 0.625487i	−0.949838	−	0.312743i	$-0.898752\pi$
$977$	− 1.86618e7i	− 0.625487i	0.949838	−	0.312743i	$-0.101248\pi$
$978$	− 2.23988e7i	− 0.748821i
$979$	2.52638e7	0.842445
$980$	0	0
$981$	685781.	0.0227516
$982$	6.65073e7i	2.20085i
$983$	− 4.07152e6i	− 0.134392i	−0.997740	−	0.0671959i	$-0.978595\pi$
$983$	− 4.07152e6i	− 0.134392i	0.997740	−	0.0671959i	$-0.0214053\pi$
$984$	1.26008e7	0.414869
$985$	0	0
$986$	295163.	0.00966873
$987$	− 3.45812e7i	− 1.12992i
$988$	7.07720e6i	0.230658i
$989$	8.48466e6	0.275832
$990$	0	0
$991$	3.55164e7	1.14880	0.574401	−	0.818574i	$-0.305235\pi$
$991$	3.55164e7	1.14880	0.574401	+	0.818574i	$0.305235\pi$
$992$	− 6.21680e7i	− 2.00580i
$993$	− 2.05943e7i	− 0.662786i
$994$	−3.91033e7	−1.25530
$995$	0	0
$996$	4.48616e6	0.143293
$997$	2.88979e7i	0.920721i	0.887732	+	0.460360i	$0.152280\pi$
$997$	2.88979e7i	0.920721i	−0.887732	+	0.460360i	$0.847720\pi$
$998$	− 6.85491e7i	− 2.17859i
$999$	1.32166e7	0.418992

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.b.h.274.5		18
5.2	odd	4		325.6.a.h.1.8	✓	9
5.3	odd	4		325.6.a.i.1.2	yes	9
5.4	even	2	inner	325.6.b.h.274.14		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.6.a.h.1.8	✓	9	5.2	odd	4
325.6.a.i.1.2	yes	9	5.3	odd	4
325.6.b.h.274.5		18	1.1	even	1	trivial
325.6.b.h.274.14		18	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	325.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-7.28147i q^{2} -14.9935i q^{3} -21.0198 q^{4} -109.175 q^{6} -135.391i q^{7} -79.9521i q^{8} +18.1955 q^{9} +O(q^{10})\)	\(q-7.28147i q^{2} -14.9935i q^{3} -21.0198 q^{4} -109.175 q^{6} -135.391i q^{7} -79.9521i q^{8} +18.1955 q^{9} +191.317 q^{11} +315.160i q^{12} +169.000i q^{13} -985.848 q^{14} -1254.80 q^{16} +874.338i q^{17} -132.490i q^{18} -1992.26 q^{19} -2029.99 q^{21} -1393.07i q^{22} -2091.74i q^{23} -1198.76 q^{24} +1230.57 q^{26} -3916.23i q^{27} +2845.90i q^{28} +46.3621 q^{29} -9450.42 q^{31} +6578.33i q^{32} -2868.52i q^{33} +6366.46 q^{34} -382.466 q^{36} +3374.82i q^{37} +14506.6i q^{38} +2533.90 q^{39} -10511.6 q^{41} +14781.3i q^{42} +4056.27i q^{43} -4021.45 q^{44} -15230.9 q^{46} +17035.2i q^{47} +18813.8i q^{48} -1523.82 q^{49} +13109.4 q^{51} -3552.34i q^{52} -23984.5i q^{53} -28515.9 q^{54} -10824.8 q^{56} +29870.9i q^{57} -337.584i q^{58} +10197.8 q^{59} -27440.0 q^{61} +68812.9i q^{62} -2463.52i q^{63} +7746.27 q^{64} -20887.0 q^{66} +317.851i q^{67} -18378.4i q^{68} -31362.5 q^{69} +39664.7 q^{71} -1454.77i q^{72} +4764.06i q^{73} +24573.7 q^{74} +41876.9 q^{76} -25902.7i q^{77} -18450.5i q^{78} +74545.9 q^{79} -54296.4 q^{81} +76539.5i q^{82} -14234.5i q^{83} +42669.9 q^{84} +29535.6 q^{86} -695.130i q^{87} -15296.2i q^{88} +132052. q^{89} +22881.1 q^{91} +43967.9i q^{92} +141695. i q^{93} +124041. q^{94} +98632.1 q^{96} -19366.7i q^{97} +11095.6i q^{98} +3481.12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9}+O(q^{10}) \) 18 * q - 182 * q^4 - 166 * q^6 - 1124 * q^9	\( 18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9} - 2844 q^{11} + 684 q^{14} - 2122 q^{16} + 816 q^{19} - 7824 q^{21} + 16938 q^{24} - 1690 q^{26} + 17474 q^{29} + 1496 q^{31} + 35578 q^{34} + 1024 q^{36} + 3718 q^{39} - 57352 q^{41} + 61198 q^{44} - 24112 q^{46} + 81814 q^{49} - 62012 q^{51} + 205522 q^{54} - 46004 q^{56} + 176284 q^{59} + 56330 q^{61} + 201690 q^{64} + 85154 q^{66} + 363494 q^{69} - 141124 q^{71} + 271352 q^{74} + 92746 q^{76} + 328146 q^{79} - 139870 q^{81} + 691312 q^{84} - 589840 q^{86} + 505396 q^{89} + 4056 q^{91} + 1003212 q^{94} - 638362 q^{96} + 1515552 q^{99}+O(q^{100}) \) 18 * q - 182 * q^4 - 166 * q^6 - 1124 * q^9 - 2844 * q^11 + 684 * q^14 - 2122 * q^16 + 816 * q^19 - 7824 * q^21 + 16938 * q^24 - 1690 * q^26 + 17474 * q^29 + 1496 * q^31 + 35578 * q^34 + 1024 * q^36 + 3718 * q^39 - 57352 * q^41 + 61198 * q^44 - 24112 * q^46 + 81814 * q^49 - 62012 * q^51 + 205522 * q^54 - 46004 * q^56 + 176284 * q^59 + 56330 * q^61 + 201690 * q^64 + 85154 * q^66 + 363494 * q^69 - 141124 * q^71 + 271352 * q^74 + 92746 * q^76 + 328146 * q^79 - 139870 * q^81 + 691312 * q^84 - 589840 * q^86 + 505396 * q^89 + 4056 * q^91 + 1003212 * q^94 - 638362 * q^96 + 1515552 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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